Lomonosov's invariant subspace theorem is a mathematical theorem from functional analysis concerning the existence of invariant subspaces of a linear operator on some complex Banach space. The theorem was proved in 1973 by the Russian–American mathematician Victor Lomonosov.
Lomonosov's invariant subspace theorem
Notation and terminology Let B ( X ) := B ( X , X ) {\displaystyle {\mathcal {B}}(X):={\mathcal {B}}(X,X)} be the space of bounded linear operators from some space X {\displaystyle X} to itself. For an operator T ∈ B ( X ) {\displaystyle T\in {\mathcal {B}}(X)} we call a closed subspace M ⊂ X , M ≠ { 0 } {\displaystyle M\subset X,\;M\neq \{0\}} an invariant subspace if T ( M ) ⊂ M {\displaystyle T(M)\subset M} , i.e. T x ∈ M {\displaystyle Tx\in M} for every x ∈ M {\displaystyle x\in M} .
Theorem Let X {\displaystyle X} be an infinite dimensional complex Banach space, T ∈ B ( X ) {\displaystyle T\in {\mathcal {B}}(X)} be compact and such that T ≠ 0 {\displaystyle T\neq 0} . Further let S ∈ B ( X ) {\displaystyle S\in {\mathcal {B}}(X)} be an operator that commutes with T {\displaystyle T} . Then there exist an invariant subspace M {\displaystyle M} of the operator S {\displaystyle S} , i.e. S ( M ) ⊂ M {\displaystyle S(M)\subset M} .
Citations
References Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
